Symmetric closure

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In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X that contains R.

For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y", then the symmetric closure of R is the relation "there is a direct flight either from x to y or from y to x". Or, if X is the set of humans (alive or dead) and R is the relation 'parent of', then the symmetric closure of R is the relation "x is a parent or a child of y".

Definition[edit]

The symmetric closure S of a relation R on a set X is given by

In other words, the symmetric closure of R is the union of R with its inverse relation, R -1.

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